Combinatorial Groups

I have tried to write an equivalent definition of a combinatorial group, inspired by the Burnside definition of a transitive group.

Burnside wrote:

133. Definition. A permutation-group is called transitive when, by means of its permutations, a given symbol a₁ can be changed into every other symbol a₂, a₃, ..., a_n operated on by the group. When it has not this property, the group is called intransitive.

Here is what I got :

Equivalent definition
Let H be a closed subset of n permutations of the symmetric group Sn that permutes the symbols a₁,......, a_n.
H is called combinatorial group if H is transitive, i.e, by means of its permutations, a given symbol a₁ can be changed into every other symbol a₁, a₂, a₃, ..., a_n operated by H.

Example :


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The next piece that I need to explain why GroupeCombinatoire' = Lin is the Big Bijection Theory.

To understand the Big Bijection Theory one must understand the Relativity Theory.

Mainly, the Relativity Theory says that the time flows with the same speed all over*, and, yes, we can synchronize clocks for two events to be linearly compared, even that a direct link cause-effect is not possible to establish between them.

Thus, every box/slot/cube/ball/label ever produced has two simultaneous attributes :

- a unique ID number (a name)
- a unique time-stamp, that can be always compared, by the theory of relativity, with others time-stamps.

The Big Bijection Theory sustains that there is an Unique Original Bijection between the IDs of objects and their time-stamps.


Thus, a named-slots structure is the very same combinatorial thing with a linear order structure.

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* excepting, for academic rigor, accelerated environments such as roller coasters or discos.

nick wrote:

the Big Bijection Theory

Hey bro, do you ever heard about the Big Bijection Theory ?

Nick, just for academic honesty, why don't you tell fellows where do you saw the very first time the combinatorial group
?

Dig Da Link, bros,
Dig Da Link !

=======================
Look here.

There are people that are studying 4D synthetic geometry. One day, someone will write a computer program that will train the user to develop his 4D intuition. Some stunt man will wire himself to the computer, and he will obtain the 4D intuition.

I think that these kind of intuition's experiments must come in a natural manner. If I will ever understand a combinatorial identity like

a Normal Subgroup = a Group ( a Group)

or whatever is a group labeled with another group, this must come naturally, like the snow in a beautiful winter as this one.

Nick, this is no more about sets and/or arrows.

This is about Graphs and/or Groups.

After you point a node in a graph, you get a set of neighbors.
After you point a node in a group, you get a lin of neighbors.

Dr. Post wrote:

After you point a node in a graph, you get a set of neighbors.
After you point a node in a group, you get a lin of neighbors.

That's exactly what I am saying. Moreover,

After I point a node in a cube, I get a cyc of neighbors.

a cyc from Cyclops, not from bananas or strawberries.

Nick, here is one good news : The century is XXI ! here is is 12-puzzle !

http://magnolia-heights.com/sporadicgames/m12/manual.html

Dr. Post wrote:

The century is XXI !

So what ?
It's on UQAM labels factory to produce nice labels, like group-labels.

Nick, let's bypass bla bla. and let me ask you something,

If you ever need to quote about UQAM species,
would you use

Joyal ?
or
Joyal of Saint-Majorique ?

Who wrote the very first time Subset = Set.Set?

Doc, this is about the natural generalization of Cardinal Numbers and Ordinals Numbers.

Normally I use UQAM teachers and not Joyal of Saint-Majorique .

you know,

this natural generalization of Cardinal Numbers and Ordinal Numbers exists only in your imagination, like all Math in the world.
Math does not exists in books, but in the heads of people that claim them selves mathematicians.


Until another claim, you should use "Nick of Longueuil" to point who did this generalization.

My maximal claim is to be the very first student that understood the Theory of Species.

Until another claim, Group' remains Lin, via the Big Bijection Theory.

After you point a node in a graph, you get a set of neighbors,
After you point a node in a group, you get a lin of neighbors,
After you point a node in a cube, you a cyc of neighbors, and
After you point a node in a function, you get an X of neighbors.
==============
Let's stick on subject :

Nick, on this planet, Z2×Z2 is :

{111, 1ii, 1jj, 1kk, i1i, j1j, k1k, ii1, jj1, kk1, ijk, jik, ikj, kij, kji, jki }

When labeling, you have one special element and four choices for the first label. To label the rest, is to label { i, j, k, ijk, jik, ikj, kiji, kji, jki } that is an Ens3.

all over you get X.Ens3;

(X.Ens3)' = Ens3 + X.Ens2

My combinatorial Z2×Z2 is :

{ 1ijk, i1kj, jk1i, kji1 }

(Z2× Z2)' = Lin3

There are six realizations and not four.
The Stab is the group itself Z2×Z2 and not Aut(Z2×Z2 ).

nick wrote:

the Big Bijection Theory.
Thus, every box/slot/cube/ball/label ever produced has two simultaneous attributes :
- a unique ID number (a name)
- a unique time-stamp, that can be always compared, by the theory of relativity, with others time-stamps.

The only solution I see is to develop the Big Bijection Theory.

Thus, every abstract group has two attributes :

- a Burnside group ( primitive of some lin )
- a Cayley group ( or Cayley table )

Marry Christmas everyone !

G funk wrote:

Nick, on this planet, Z2×Z2 is :
{ 111, 1ii, 1jj, 1kk, i1i, j1j, k1k, ii1, jj1, kk1, ijk, jik, ikj, kij, kji, jki }

And what happens if an extraterrestrial brother reads the set above ?
Do you think that it will verify the axioms ? No !

It will say : here is the Aut (Z2×Z2) !

Ok.

Let's take Frobenius of order 20 with degree 5, sharply double transitive.
ID.........12345
[1, 2, 3, 4, 5]...23451
[1, 3, 5, 2, 4]...34512
[1, 4, 2, 5, 3]...45123
[1, 5, 4, 3, 2]...51234

[2, 5], [3, 4]...15432
[1, 3], [4, 5]...32154
[1, 5], [2, 4]...54321
[1, 2], [3, 5]...21543
[1, 4], [2, 3]...43215

[2, 3, 5, 4]...13524
[2, 4, 5, 3]...14253

[1, 4, 3, 5]...42531
[1, 5, 3, 4]...52413

[1, 2, 5, 4]...25314
[1, 4, 5, 2]...41352

[1, 3, 2, 5]...35241
[1, 5, 2, 3]...53142

[1, 2, 4, 3]...24135
[1, 3, 4, 2]...31425

and let's take the derivative :
*2345
2345*
345*2
45*23
5*234

*5432
32*54
5432*
2*543
432*5

*3524
*4253

4253*
524*3

253*4
4*352

3524*
53*42

24*35
3*425

A close look reveals that da stabilizer of this structure is ={ (), (2453), (25)(34), (2354) }= Z4, or C4.

Doc, doesn't this looks like F₅'=Z₄ ?
and, generally speaking,

Field'=Cyc ?

Nick, we give up ! Here is the correct formula for your cube :

CUBE' = X. [Cyc₄(X⁶) + Cyc₃(X⁸) + Cyc₂(X¹²)]

After pinning a special point, another point becomes very special.
The remaining liberty of moving (relabeling) is described by
- either a Cyc₄(X⁶)
- either a Cyc₃(X⁸)
- or a Cyc₂(X¹²) (no coefficients are needed), where aCyc(aLin) expresses the solidity.
The e.g.f. is x(1/4+1/3+1/2)x^24 hence you have 13/12.25! = 26!/24 distinct labellings (with 26 distinct labels)

----
So, after pinning a point in aField, everybody will believe that the pinned point is 0, and the remaining liberty is aCyc one...

Who's next ? do you have an example of endo-species aSpecies that has no primitive ?

OK. For the reader that tries to follow these posts, here is the dictionary :

transposition canon

The nice photos were taken with a solar system of transpositions: (0, 1), (0, 2),......,(0, n-1).

What I (at least) have claimed is :

E± = Taquin
P₄^bic = Z₂×Z₂ and aGroup'= aLin
P₅/Z₂ = F₅ and aField' = aCyc

Take one from me :

Since for each n there is precisely one Boolean Algebra, we must have :

BooleanAlgebra = Ens

Doc is right, Nick.

powerset (A), A×A, A^A are in the same situation.

The secret of trees is that those nodes have a very small amount of structure around them; the trees are the most "naked" structures among others.

A tree acting on its nodes is equal to the tree of edges (with concatenation).
They are very complex and simultaneously exposed structure, hence their generosity as species.

To compare, take a look here to see the mess inside a finite vector space : http://math.illinoisstate.edu/psissok/publ/art3.pdf

Yep. To integrate a field, one must add a point (namely infinity) and consider the Möbius transformations - that form a sharply triple transitive group.

Möbius Fields -----> Affine Fields -----> Cycs, Groups ----- >Lins That's the way aha aha !


Fellows, my inquiry comes to its end... as doctor said, we can not find new things about boolean algebras using the four basic operations of species (of numbers).

When passing to species we loose axioms, constants, relations, operations: plenty of useful things that make math. Nevertheless, an object caught in a simple combinatorial equation must be a fundamental math object that deserves to be studied.

Again good news !
I have found new tables of species.

It is no more about derivation, but is about disintegration !
Probably, the word "derivation" was used to help the student to remember what he has to do when passing to e.g.f.

http://www.labmath.uqam.ca/~annales/volumes/17-1/PDF/011-037.pdf
Hence :

(Z₆/A₅b)/Z₂ ----disintegration---> P₅/Z₂ ----disintegration---> C₄ ----disintegration---> L₃

A Möbius Field disintegrates into an Affine Field that disintegrates into a Cyclic Group that disintegrates into a Lin.

http://home.wlu.edu/~dresdeng/smallrings/
Other good news is that there are intermediary objects between groups and fields, that could be rings.
They are more than sharply transitive and less that sharply double transitive.

Again, let's take C₇, P₇, C₇/Z₃ and P₇/Z₃ from tables.

meaning respectively group (Z₇) , some ring, some ring, and field (F₇)

One can see those species as linear transformations a.x+b over Z₇, where

- a belongs to {1}
- a belongs to {1, 6}
- a belongs to {1, 2, 4} or
- a belongs to {1, 2, 3, 4, 5, 6} respectively.

And here is the new inquiry.

Once accepted the identity-free representation of a permutation group (a closed subset of permutations),
e.g. the Diamond {12abc, 12bca, 12cab, 21bac, 21cba, 21acb} has order 6 and degree 5,

also written as a rectangle
12abc
12bca
12cab
21bac
21cba
21acb
that works since anyone of these xyztu may be appointed as identity,

the four basic operations with species become :

Sum :
Means just to put nearby two rectangles,

Product :
- the new degree is the sum of the degrees
- the lines of the new rectangle are the all concatenations of the lines of the two tables

Disintegration :
Means to reduce the degree with one and to clean up the messed rectangle(s)
new degree = degree - 1
new order' = order/degree

Composition :
It is the wreath product
Means to substitute the symbols in the first rectangle with lines and copies of lines of the second rectangle.
new degree = degree1×degree2
new order = order1×(order2)^degree1

Hall (1959) describes the composition (wreath product).

Ok.

so, a species is no more some clouds or traines des nuages in the blue sky that suffers magical interferences with other clouds, but is a permutation group (or a series of).

Let's call this group a ground group.

Nick, will you explain :

- why the ground group of a group G is the group acting on itself and not Aut(G)
- why the ground group of a field F is not (F,+), not (F, ×) but (F, ax+b)

etc etc. Does a math structure has a unique ground group and how do you obtain it ?

Fellows,

Didn't happened to you to hear people comparing math with poetry or music ?

Well, this is what they were talking about : to find the ground group(s) of some given structure.